Theorem iunsuc 4405

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
iunsuc.1	|- A e. _V
iunsuc.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunsuc	|- U_ x e. suc A B = ( U_ x e. A B u. C )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 4356	. . 3 |- suc A = ( A u. { A } )
2		iuneq1 3886	. . 3 |- ( suc A = ( A u. { A } ) -> U_ x e. suc A B = U_ x e. ( A u. { A } ) B )
3	1, 2	ax-mp 5	. 2 |- U_ x e. suc A B = U_ x e. ( A u. { A } ) B
4		iunxun 3952	. 2 |- U_ x e. ( A u. { A } ) B = ( U_ x e. A B u. U_ x e. { A } B )
5		iunsuc.1	. . . 4 |- A e. _V
6		iunsuc.2	. . . 4 |- ( x = A -> B = C )
7	5, 6	iunxsn 3949	. . 3 |- U_ x e. { A } B = C
8	7	uneq2i 3278	. 2 |- ( U_ x e. A B u. U_ x e. { A } B ) = ( U_ x e. A B u. C )
9	3, 4, 8	3eqtri 2195	1 |- U_ x e. suc A B = ( U_ x e. A B u. C )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   {csn 3583   U_ciun 3873   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-iun 3875  df-suc 4356

This theorem is referenced by: (None)